There exist a large family of noncommutative spaces that arise from the quantum matrices. These algebraic objects $q$-deform the coordinate rings of certain varieties. For example, take quantum $SU(2)$, this is the algebra $< a,b,c,d >$ quotiented by the ideal generated by
$$
ab−qba, ~~ ac−qca, ~~ bc−cb, ~~ bd−qdb, ~~ cd−qdc, ~~ ad−da−(q−q^{−1})bc,
$$
and the "q-det" relation
$$
ad−qbc−1
$$
where $q$ is some complex number. Clearly, when $q=1$ we get back the coordinate ring of $SU(2)$. In the classical case $S^2 = SU(2)/U(1)$ (the famous Hopf fibration). This generalises to the q-case: the $U(1)$-action generalises to a $U(1)$-coaction with an invariant subalgebra that q-deforms the coordinate algebra of $S^2$ - the famous Podles sphere. There exist such q-matrix deformations of all flag manifolds.

Since all such manifolds are Kahler, we can also apply Kontsevich deformation to them to obtain a q-defomation. My question is: What is the relationship between these two approaches?

Alternatively, we can apply Kostant-Souriau geometric quantization to a flag manifold. How does alegbra relate to its q-matrix deformation?

4 Answers
4

As far as I understand, the flag manifolds with Kahler structures mentioned in the question are simply coadjoint orbits of compact Lie groups with the Kirillov-Kostant-Souriau bracket, so their quantizations will yield quotients of the usual enveloping algebra $U(g)$ and will not have to do with quantum groups. I suppose that the q-spaces discussed in the question are meant to be q-deformations of these.

Here are some papers about this: arXiv:math/0206049 and arXiv:math/9807159. This is a rather subtle business: e.g., it is explained that the 2-parameter deformations (similar to the 2-parameter family of Podles spheres) do not always exist, although they do exist in type A
and in many other cases, e.g. if orbits are symmetric spaces.

Let $q$ be a variable instead of a complex number, write $q=1+t$ with $t$ also a variable, and complete with respect to the $t$-adic topology. This gives you a formal deformation of the coordinate algebra of the sphere in the sense of Gerstenhaber---this is not tautological: it follows from the fact that the deformation is flat.

I had the same question, but not even for quantum flag manifolds (these are quantum homogenous spaces for quantum groups), but already for the quantum groups themselves. The latter are special quantizations of so called standard Poisson-Lie structures on the group (simply connected, simple). It's thus unclear to me the relation with the canonical quantization of Kontsevich.